Prove that the series SUM (-1)^n n/p_n converges where p_n are primes

[Dragonfall]

Homework Statement

Prove that $$\sum(-1)^n\frac{n}{p_n}$$ converges, where $$p_n$$ is the nth prime.

Homework Equations

The sequence $$\frac{n}{p_n}$$ is definitely not monotone if there exists infinitely many twin primes, since $$2n-p_n<0$$ for sufficiently large n, so alternating series test is out. Are there any other ways of showing this converges?

 

[StatusX]

Can you use the prime number theorem? This says that:

$$\lim_{n \rightarrow \infty} \frac{p_n}{n \ln n} = 1$$

 

[Dragonfall]

I can't use it for the series. I can only establish that n/p_n -> 0, which is insufficient for the series. I can't even prove that n/p_n is NOT monotone for large n, unless I assume the twin prime conjecture, for example.